library(tidyverse)
library(here)
library(knitr)
library(gganimate)
library(broom)
# loading in our data sets
gdp <- read_csv(here("data", "gdp_pcap.csv"))
wateraccess <- read_csv(here("data", "at_least_basic_water_source_overall_access_percent.csv"))For this project, we are interested in assessing the relationship between a country’s GDP and the percentage of people within that country that have access to a basic water source. To do accomplish this, we will be using two datasets from Gapminder that correspond to our variables of interest.
Our two variables of interest (and their related datasets) are:
At least a basic water source, overall access % [Response]
This variable’s dataset displays the percent of people who are using at least basic water services by Country and by Year from 2000 to 2022. The percentage encompasses both the people who have access to basic water services as well as those who have access to safe water services. The dataset defines a basic water service as water from an improved source and collection is not more than 30 minutes for a round trip.
GDP per capita (Price and inflation adjusted) [Explanatory]
This variable’s dataset displays the Gross domestic product per person adjusted for differences in purchasing power, where each observation displays information for a Country in a given Year beginning in 1800 and going to present day. The values are displayed in international dollars and are fixed at 2017 prices.
We hypothesize that there will be a positive relationship between the percentage of the country’s population that has access to water and the GDP per capita. This means that as the percentage of the population has access to water increases, then the GDP per capita of the country will also increase. Our basis for this is that countries with more prevalent global economies such as the United States and European countries have better access to water and safer modern ways for their citizens to access water. And when people have better access to water they can spend more time developing other areas of the country and business which leads to a more powerful economy and a better GDP per capita.
Because each variable has their own dataset, we will have to clean and combine the two datasets together to assess the relationship between the two variables. Starting with the GDP per capita dataset, some values have the letter k embedded alongside the numeric value, preventing us to pivot the dataset into a longer format. To get around this, we forced every column in our dataset to become character variables so every column is the same data type. Once our data has been pivoted, we could then address the issue of embedded k’s by extracting the numbers for values that did contain a k, and then multiplying these values by 1000 while also converting these values to a numeric data type. Afterwards, we converted the remaining variables (country and year) to an appropriate data type.
The process for cleaning the water access dataset was much easier than for the GDP per capita dataset, as we did not have to deal with the issue of embedded k’s. Therefore, all we had to do for this dataset was pivot and then convert the remaining columns/variables to an appropriate data type. Additionally, we dropped any observtions that contained any missing or NA values.
Once each dataset had been pivoted and cleaned, we could then join our two datasets through an inner join by country and by year. To help with visualization, we also created a new variable called region which indicates the world region that a country resides in.
# Pivoting and cleaning the gdp dataset
gdp_clean <- gdp |>
mutate(across(.cols = everything(),
.f = ~ as.character(.x)
)
) |>
pivot_longer(cols = `1800`:`2100`,
names_to = "year",
values_to = "gdp_pc"
) |>
mutate(gdp_pc = if_else(condition = str_detect(gdp_pc, "k$"),
true = as.numeric(str_extract(gdp_pc, "[0-9|.]*")) * 1000,
false = as.numeric(gdp_pc)
),
country = as.factor(country),
year = as.numeric(year)
)
# Pivoting and cleaning the water_access dataset
wateraccess_clean <- wateraccess |>
pivot_longer(cols = `2000`:`2022`,
names_to = "year",
values_to = "access_pct",
values_drop_na = TRUE
) |>
mutate(country = as.factor(country),
year = as.numeric(year),
access_pct = as.numeric(access_pct)
)
# Joining the two datasets
gdp_v_water <- gdp_clean |>
inner_join(wateraccess_clean,
by = join_by(country, year)
) |>
mutate(region = fct_collapse(country,
"Asia" = c("Afghanistan",
"Australia",
"Bahrain",
"Bangladesh",
"Bhutan",
"Brunei",
"Cambodia",
"China",
"Hong Kong, China",
"Fiji",
"India",
"Indonesia",
"Iran",
"Iraq",
"Israel",
"Japan",
"Jordan",
"Kazakhstan",
"Kiribati",
"Kuwait",
"Kyrgyz Republic",
"Lao",
"Lebanon",
"Malaysia",
"Maldives",
"Marshall Islands",
"Micronesia, Fed. Sts.",
"Mongolia",
"Myanmar",
"Nauru",
"Nepal",
"New Zealand",
"North Korea",
"Oman",
"Pakistan",
"Palestine",
"Palau",
"Papua New Guinea",
"Philippines",
"Qatar",
"Samoa",
"Saudi Arabia",
"Singapore",
"Solomon Islands",
"South Korea",
"Sri Lanka",
"Syria",
"Tajikistan",
"Thailand",
"Timor-Leste",
"Tonga",
"Turkmenistan",
"Tuvalu",
"UAE",
"Uzbekistan",
"Vanuatu",
"Vietnam",
"Yemen"),
"Africa" = c("Algeria",
"Angola",
"Benin",
"Botswana",
"Burkina Faso",
"Burundi",
"Cameroon",
"Cape Verde",
"Central African Republic",
"Chad",
"Comoros",
"Congo, Dem. Rep.",
"Congo, Rep.",
"Cote d'Ivoire",
"Djibouti",
"Egypt",
"Equatorial Guinea",
"Eritrea",
"Eswatini",
"Ethiopia",
"Gabon",
"Gambia",
"Ghana",
"Guinea",
"Guinea-Bissau",
"Kenya",
"Lesotho",
"Liberia",
"Libya",
"Madagascar",
"Malawi",
"Mali",
"Mauritania",
"Mauritius",
"Morocco",
"Mozambique",
"Namibia",
"Niger",
"Nigeria",
"Palau",
"Rwanda",
"Sao Tome and Principe",
"Senegal",
"Seychelles",
"Sierra Leone",
"Somalia",
"South Africa",
"South Sudan",
"Sudan",
"Tanzania",
"Togo",
"Tunisia",
"Uganda",
"Zambia",
"Zimbabwe"),
"Americas" = c("Antigua and Barbuda",
"Argentina",
"Bahamas",
"Barbados",
"Belize",
"Bolivia",
"Brazil",
"Canada",
"Chile",
"Colombia",
"Costa Rica",
"Cuba",
"Dominica",
"Dominican Republic",
"Ecuador",
"El Salvador",
"Grenada",
"Guatemala",
"Guyana",
"Haiti",
"Honduras",
"Jamaica",
"Mexico",
"Nicaragua",
"Panama",
"Paraguay",
"Peru",
"St. Kitts and Nevis",
"St. Lucia",
"St. Vincent and the Grenadines",
"Suriname",
"Trinidad and Tobago",
"USA",
"Uruguay",
"Venezuela"),
"Europe" = c("Albania",
"Andorra",
"Armenia",
"Austria",
"Azerbaijan",
"Belarus",
"Belgium",
"Bosnia and Herzegovina",
"Bulgaria",
"Croatia",
"Cyprus",
"Czech Republic",
"Denmark",
"Estonia",
"Finland",
"France",
"Georgia",
"Germany",
"Greece",
"Hungary",
"Iceland",
"Ireland",
"Italy",
"Latvia",
"Lithuania",
"Luxembourg",
"Malta",
"Moldova",
"Monaco",
"Montenegro",
"Netherlands",
"North Macedonia",
"Norway",
"Poland",
"Portugal",
"Romania",
"Russia",
"San Marino",
"Serbia",
"Slovak Republic",
"Slovenia",
"Spain",
"Sweden",
"Switzerland",
"Turkey",
"UK",
"Ukraine"),
other_level = "other"
),
.after = country
)
kable(head(gdp_v_water), format = "markdown")| country | region | year | gdp_pc | access_pct |
|---|---|---|---|---|
| Afghanistan | Asia | 2000 | 794 | 27.4 |
| Afghanistan | Asia | 2001 | 775 | 27.5 |
| Afghanistan | Asia | 2002 | 1260 | 29.7 |
| Afghanistan | Asia | 2003 | 1280 | 31.9 |
| Afghanistan | Asia | 2004 | 1260 | 34.1 |
| Afghanistan | Asia | 2005 | 1350 | 36.3 |
Now that we have our combined dataset, we can now assess the relationship between a country’s GDP per capita and the proportion of their population that has access to water. Before formally analyzing this relationship through a fitted linear regression model, we would like to look at the relationship between the two variables over time.
gdp_v_water |>
ggplot(aes(x = gdp_pc,
y = access_pct,
fill = region,
color = region,
shape = region)
) +
geom_point() +
theme_bw() +
scale_color_manual(values = c("#e66101",
"#fdb863",
"#b2abd2",
"#5e3c99")
)+
scale_fill_manual(values = c("#e66101",
"#fdb863",
"#b2abd2",
"#5e3c99")
) +
scale_shape_manual(values = 21:24) +
labs(x = "GDP Per Capita",
y = "",
subtitle = "% of Pop. with Basic Water Access",
title = "Relationship between Water Access and GDP",
tag = "Year: {frame_time}",
fill = "World Region",
color = "World Region",
shape = "World Region"
) +
transition_time(as.integer(year)) +
ease_aes('linear')
When assessing the relationship between Water Access vs. GDP over time through an animated plot, we can see that there generally seems to be a strong, positive non-linear relationship between the two variables; while countries with larger GDP per capita tend to have higher rates of water access, this trend only seems to apply when looking at countries that have a GDP per Capita of about $50,000 or less. Countries with a GDP per capita of $50,000 or greater will more often than not have almost 100% of their population have access to a basic water source. Additionally, as time goes on, we see that there seems to be an improvement overall in the percentage of the population that have water access for all countries, with most countries having been able to increase this percentage over time. One other notable observation that can be made from this animated plot is that the world region with seemingly the highest percentage of water access on average is Europe, while the region with the lowest percentage on average seems to be Africa (which makes sense as European countries seem to have the highest GDP per capita while African countries seem to have the lowest).
Because we can see that the relationship between GDP per capita and Water Access is far from being linear, we would like to perform a logit transformation on the average proportion of a country’s population that has basic water access (our response variable) before fitting a linear model to help with this issue of non-linearity using the following formula:
ln(average water access / (101 - average water access))
A plot containing a fitted linear model alongside our transformed response variable follows.
gdp_v_water_mean <- gdp_v_water |>
group_by(country, region) |>
summarize(mean_gdp = mean(gdp_pc),
mean_access = mean(access_pct)
) |>
mutate(logit_meanaccess = log((mean_access)/(101 - mean_access)))
gdp_v_water_mean |>
ggplot(aes(x = mean_gdp,
y = logit_meanaccess
)
) +
geom_point() +
geom_smooth(method = "lm",
aes(x = mean_gdp,
y = logit_meanaccess,
group = 1),
color = "red"
) +
theme_bw() +
labs(x = "GDP Per Capita (averaged over the years)",
y = "",
subtitle = "Logit of the Pop. Percentage with Basic Water Access (averaged over the years)",
title = "Relationship between LOGIT avg. Water Access and avg. GDP"
) +
scale_y_continuous(limits = c(0, 6))
From this plot, we can see that even with a transformation, there still seems to be a somewhat weak, positive linear relationship between the average GDP per capita of a country and the logit of the population percentage with access to a basic water source, though this relationship is much improved in terms of linearity over the relationship with the original non transformed response. Additionally, we can also see that there may be some data points in which some people may consider outliers. Most notably, there is one country that has a relatively high GDP per capita compared to the other countries. By rearranging our combined dataset by the mean GDP per capita in descending order and grabbing the first observation, we can determine that this country with the highest average GDP per capita is Monaco, a country in Europe with an average GDP per capita of $186,521.74 and a logit value of 4.6052.
gdp_v_water_mean |>
ungroup() |>
slice_max(order_by = mean_gdp,
n = 1)In addition to our plot containing a linear regression model, we can also create and obtain the coefficients for this model.
model <- lm(logit_meanaccess ~ mean_gdp, data = gdp_v_water_mean)
model
Call:
lm(formula = logit_meanaccess ~ mean_gdp, data = gdp_v_water_mean)
Coefficients:
(Intercept) mean_gdp
1.526e+00 4.629e-05 #summary(model)
#augment(model)
#anova(model)Our estimated linear regression model (after performing a logit transformation on our response variable) is:
ŷ = predicted ln(mean water access/(101- mean water access)) = 1.516 + 0.0000463(mean GDP per capita).
Since the intercept for our model is 1.516, then the logit value for countries with an average GDP per capita of zero is 1.516 (corresponding to a population percentage value of 82.83, which was obtained by undoing the original logit transformation through an inverse). Similarly, because our model’s slope is 0.0000463, then when a country’s GDP per capita increases by one dollar, the logit value of the population percentage that has access to a basic water supply is expected to increase by 0.0000463.
(INCLUDE ASSESSMENT OF LINE CONDITIONS HERE)
kable(
tibble(LogitAccess_Variance = var(augment(model)$logit_meanaccess),
Prediction_Variance = var(augment(model)$.fitted),
Residual_Variance = var(augment(model)$.resid)
)
)| LogitAccess_Variance | Prediction_Variance | Residual_Variance |
|---|---|---|
| 2.454607 | 1.156303 | 1.298303 |
To assess our model’s fit, we can use variances for the response, fitted, and residual values. From here, we can calculate the proportion of the variability seen in our response variable that can be explained by our estimated linear regression model, which in this case is:
1.156/2.455 = 1-(1.300/2.455) = 0.471
Only about 47.1% of the variation in the response can be accounted for by our model, which indicates that we have a very poor model, and suggests that there is a very weak linear relationship between the GDP per capita of a country and the logit value of the percentage of the country’s population that has access to a basic water source. Therefore, a linear model may not be the best way to describe the relationship between these two variables, which confirms what we saw earlier when plotting the linear regression model.
To better assess how our fitted linear model performs at explaining the total variability seen in the response variable, we can do some simulation!
To start off, we can generate simulated observations by using the predicted values from our linear regression model and add random errors to each one using the residual standard error from the same model.
set.seed(7256)
# Creating function that adds random noise to simulate 'observed' values
addnoise <- function(x, mean = 0, sd) {
new_x <- x + rnorm(length(x),
mean,
sd
)
return(new_x)
}
# Extracting predicted values and standard deviation of residuals from fitted model
logit_predict <- predict(model)
resid_sigma <- sigma(model)
# Testing function and creating a set of simulated observations
sim_logit <- tibble(sim_logit_meanaccess = addnoise(x = logit_predict, sd = resid_sigma))
sim_logit# A tibble: 194 × 1
sim_logit_meanaccess
<dbl>
1 3.03
2 2.53
3 0.406
4 4.25
5 1.54
6 2.88
7 1.89
8 1.53
9 4.00
10 1.61
# ℹ 184 more rows# Adding newly generated observations to a dataset containing the original observed values
sim_data <- gdp_v_water_mean |>
ungroup() |>
filter(is.na(logit_meanaccess) == FALSE,
is.na(mean_access) == FALSE
) |>
select(mean_gdp, logit_meanaccess) |>
bind_cols(sim_logit)
# Visualizing and comparing the simulation from our model to the original data
sim_data |>
ggplot(aes(x = mean_gdp,
y = logit_meanaccess
)
) +
geom_point() +
geom_smooth(method = "lm",
aes(x = mean_gdp,
y = logit_meanaccess,
group = 1),
color = "red"
) +
theme_bw() +
labs(x = "GDP Per Capita (averaged over the years)",
y = "",
subtitle = "Logit of the Pop. Percentage with Basic Water Access (averaged over the years)",
title = "Relationship between Observed LOGIT avg. Water Access and avg. GDP"
) +
scale_y_continuous(limits = c(0, 6))
sim_data |>
ggplot(aes(x = mean_gdp,
y = sim_logit_meanaccess
)
) +
geom_point() +
geom_smooth(method = "lm",
aes(x = mean_gdp,
y = logit_meanaccess,
group = 1),
color = "red"
) +
theme_bw() +
labs(x = "GDP Per Capita (averaged over the years)",
y = "",
subtitle = "Simulated Logit of the Pop. Percentage with Basic Water Access (averaged over the years)",
title = "Relationship between Simulated LOGIT avg. Water Access and avg. GDP"
) +
scale_y_continuous(limits = c(0, 6))
sim_data |>
ggplot(aes(x = sim_logit_meanaccess,
y = logit_meanaccess),
) +
geom_point() +
scale_x_continuous(limits = c(0, 10)) +
scale_y_continuous(limits = c(0, 10)) +
geom_abline(slope = 1, intercept = 0, color = "red", linewidth = 1) +
theme_bw() +
labs(x = "Simulated Logit of the Pop. Percentage with Basic Water Access (averaged over the years)",
y = "",
subtitle = "Observed Logit of the Pop. Percentage with Basic Water Access (averaged over the years)",
title = "Relationship between Simulated LOGIT avg. Water Access and Observed LOGIT avg. Water Access"
)
Instead of creating just one dataset of simulated observations, we can generate 1000 simulated datasets. From here, we can regress the observed dataset against each one of the 1000 simulated datasets to generate multiple R^2 values, from where we can assess the distribution of the R^2 values to determine how well our original linear regression model describes the dataset (by determining how much variability seen in the observed logit water access is accounted for by our similated data based on the linear model).
set.seed(7256)
# Generating 1000 simulated sets of observations and putting them into one dataset
nsims <- 1000
sim_data2 <- map_dfc(.x = 1:nsims,
.f = ~ tibble(sim = addnoise(logit_predict,
sd = resid_sigma
)
)
)
# Cleaning the simulated dataset
colnames(sim_data2) <- colnames(sim_data2) |>
str_replace(pattern = "\\.\\.\\.",
replace = "_"
)
# Combining this simulated dataset with a dataset containing the original observed values
sim_data2 <- gdp_v_water_mean |>
ungroup() |>
filter(is.na(mean_gdp) == FALSE,
is.na(logit_meanaccess) == FALSE
) |>
select(logit_meanaccess) |>
bind_cols(sim_data2)
head(sim_data)# A tibble: 6 × 3
mean_gdp logit_meanaccess sim_logit_meanaccess
<dbl> <dbl> <dbl>
1 1688. 0.0770 3.03
2 10617. 2.26 2.53
3 10891. 2.38 0.406
4 57026. 4.61 4.25
5 6690. 0.0102 1.54
6 19900 3.63 2.88 # Regressing observed values to each set of simulated values to generate r-squared values
reg_sim <- map(.x = sim_data2,
.f = ~ lm(logit_meanaccess ~ .x,
data = sim_data2)
) |>
map(.f = ~ glance(.x)) |>
map_dbl(.f = ~ .x$r.squared)
# Removing the r-squared value corresponding to logit_meanaccess ~ logitmeanaccess
reg_sim <- reg_sim[names(reg_sim) != "logit_meanaccess"]
reg_sim sim_1 sim_2 sim_3 sim_4 sim_5 sim_6 sim_7
0.20914196 0.21148541 0.25847887 0.25228309 0.26960882 0.17772895 0.17757995
sim_8 sim_9 sim_10 sim_11 sim_12 sim_13 sim_14
0.21894337 0.19532884 0.13547097 0.25003140 0.22454559 0.18874008 0.20815972
sim_15 sim_16 sim_17 sim_18 sim_19 sim_20 sim_21
0.17002736 0.21154163 0.25035802 0.24931050 0.17389394 0.30385721 0.20886826
sim_22 sim_23 sim_24 sim_25 sim_26 sim_27 sim_28
0.28991526 0.26896820 0.20109517 0.27030517 0.17367035 0.20962194 0.21279009
sim_29 sim_30 sim_31 sim_32 sim_33 sim_34 sim_35
0.21490715 0.27338537 0.24350575 0.26032701 0.27104741 0.34144762 0.21017809
sim_36 sim_37 sim_38 sim_39 sim_40 sim_41 sim_42
0.21664747 0.15213649 0.32810780 0.19737834 0.19050847 0.25647491 0.22450223
sim_43 sim_44 sim_45 sim_46 sim_47 sim_48 sim_49
0.27299961 0.14824886 0.19854595 0.23568179 0.21555423 0.27690730 0.25458909
sim_50 sim_51 sim_52 sim_53 sim_54 sim_55 sim_56
0.25006720 0.12201218 0.15487391 0.21437008 0.30883319 0.19921507 0.23440312
sim_57 sim_58 sim_59 sim_60 sim_61 sim_62 sim_63
0.17031740 0.27545673 0.24193946 0.23383554 0.16169791 0.33044136 0.23285774
sim_64 sim_65 sim_66 sim_67 sim_68 sim_69 sim_70
0.28654331 0.28840355 0.22428568 0.29760196 0.19003882 0.19452967 0.26087991
sim_71 sim_72 sim_73 sim_74 sim_75 sim_76 sim_77
0.15527300 0.18205842 0.20899036 0.30198594 0.24463499 0.25147532 0.26776938
sim_78 sim_79 sim_80 sim_81 sim_82 sim_83 sim_84
0.17385353 0.29640094 0.23350068 0.16686470 0.19611869 0.21006616 0.22712173
sim_85 sim_86 sim_87 sim_88 sim_89 sim_90 sim_91
0.22747449 0.24432338 0.18933972 0.20576064 0.24514067 0.23442927 0.19190672
sim_92 sim_93 sim_94 sim_95 sim_96 sim_97 sim_98
0.16070229 0.21801826 0.36018350 0.21864354 0.26860461 0.15831128 0.22369545
sim_99 sim_100 sim_101 sim_102 sim_103 sim_104 sim_105
0.20036270 0.25745066 0.19497777 0.14809788 0.23170933 0.21038727 0.26293569
sim_106 sim_107 sim_108 sim_109 sim_110 sim_111 sim_112
0.20451144 0.17995960 0.14804465 0.23002695 0.24173321 0.20336126 0.21497299
sim_113 sim_114 sim_115 sim_116 sim_117 sim_118 sim_119
0.18983563 0.26762819 0.20087707 0.19080149 0.21603075 0.19708405 0.26295139
sim_120 sim_121 sim_122 sim_123 sim_124 sim_125 sim_126
0.17303994 0.30004095 0.21240491 0.21738655 0.24440777 0.22619239 0.20958896
sim_127 sim_128 sim_129 sim_130 sim_131 sim_132 sim_133
0.19273356 0.22130061 0.21364646 0.18668722 0.21965130 0.25853841 0.23037424
sim_134 sim_135 sim_136 sim_137 sim_138 sim_139 sim_140
0.22213169 0.17204489 0.19248219 0.21809574 0.27696001 0.20558249 0.13198514
sim_141 sim_142 sim_143 sim_144 sim_145 sim_146 sim_147
0.21904652 0.16326800 0.25310232 0.20891317 0.22268855 0.18677023 0.26188900
sim_148 sim_149 sim_150 sim_151 sim_152 sim_153 sim_154
0.26587053 0.19469913 0.18952494 0.16205197 0.20827861 0.18443565 0.22789281
sim_155 sim_156 sim_157 sim_158 sim_159 sim_160 sim_161
0.21633946 0.15984873 0.19983307 0.23347338 0.13385194 0.23670478 0.20621922
sim_162 sim_163 sim_164 sim_165 sim_166 sim_167 sim_168
0.22929541 0.30900289 0.10087519 0.24966620 0.27227409 0.16113291 0.21781188
sim_169 sim_170 sim_171 sim_172 sim_173 sim_174 sim_175
0.23765895 0.21142666 0.26245543 0.25959391 0.21549365 0.16671325 0.25381154
sim_176 sim_177 sim_178 sim_179 sim_180 sim_181 sim_182
0.21392352 0.26040516 0.28008312 0.23572388 0.27682448 0.11624340 0.14670289
sim_183 sim_184 sim_185 sim_186 sim_187 sim_188 sim_189
0.24295880 0.23476743 0.27453920 0.21444859 0.31198825 0.24145577 0.35317707
sim_190 sim_191 sim_192 sim_193 sim_194 sim_195 sim_196
0.25413367 0.23798650 0.23942608 0.20427757 0.27464938 0.25864542 0.24163255
sim_197 sim_198 sim_199 sim_200 sim_201 sim_202 sim_203
0.27686065 0.19089643 0.21983970 0.27632530 0.21548867 0.24881317 0.25721697
sim_204 sim_205 sim_206 sim_207 sim_208 sim_209 sim_210
0.20341268 0.21208589 0.22963029 0.21717243 0.20983351 0.22659749 0.24710228
sim_211 sim_212 sim_213 sim_214 sim_215 sim_216 sim_217
0.29994130 0.23261132 0.30098772 0.26901904 0.20887323 0.22100304 0.24235343
sim_218 sim_219 sim_220 sim_221 sim_222 sim_223 sim_224
0.16288513 0.17214863 0.24508702 0.25717298 0.14507648 0.17157872 0.20748395
sim_225 sim_226 sim_227 sim_228 sim_229 sim_230 sim_231
0.13431556 0.22452655 0.24964557 0.27380808 0.24566942 0.29362701 0.17785672
sim_232 sim_233 sim_234 sim_235 sim_236 sim_237 sim_238
0.20060853 0.19003410 0.23388150 0.28524407 0.19126659 0.25699479 0.20741218
sim_239 sim_240 sim_241 sim_242 sim_243 sim_244 sim_245
0.22638545 0.29631376 0.23025767 0.34432582 0.26426616 0.17785540 0.25853536
sim_246 sim_247 sim_248 sim_249 sim_250 sim_251 sim_252
0.21264779 0.22904353 0.25121659 0.13842196 0.15110881 0.30317506 0.23869385
sim_253 sim_254 sim_255 sim_256 sim_257 sim_258 sim_259
0.18849731 0.19478197 0.22173259 0.19248345 0.23357685 0.24545253 0.19248510
sim_260 sim_261 sim_262 sim_263 sim_264 sim_265 sim_266
0.21917728 0.19907758 0.29012598 0.24042572 0.21514459 0.23828014 0.28537436
sim_267 sim_268 sim_269 sim_270 sim_271 sim_272 sim_273
0.24413818 0.17589158 0.28209756 0.23210312 0.19844405 0.22157579 0.23924542
sim_274 sim_275 sim_276 sim_277 sim_278 sim_279 sim_280
0.22207602 0.22341186 0.21939210 0.19810072 0.25474164 0.21757246 0.19621466
sim_281 sim_282 sim_283 sim_284 sim_285 sim_286 sim_287
0.26559166 0.21209385 0.17546724 0.14466191 0.16446428 0.27100249 0.19108489
sim_288 sim_289 sim_290 sim_291 sim_292 sim_293 sim_294
0.26155570 0.27818990 0.16776433 0.23408274 0.24122019 0.22390665 0.24080079
sim_295 sim_296 sim_297 sim_298 sim_299 sim_300 sim_301
0.20885425 0.27332102 0.24748059 0.26529855 0.19199654 0.22189056 0.19596380
sim_302 sim_303 sim_304 sim_305 sim_306 sim_307 sim_308
0.25064184 0.20818114 0.21491100 0.22810948 0.23091815 0.29680053 0.22555809
sim_309 sim_310 sim_311 sim_312 sim_313 sim_314 sim_315
0.22331345 0.20423422 0.22914119 0.19851757 0.26899864 0.21846484 0.24054408
sim_316 sim_317 sim_318 sim_319 sim_320 sim_321 sim_322
0.30065208 0.24726231 0.22003254 0.26261286 0.21739507 0.15070034 0.20874193
sim_323 sim_324 sim_325 sim_326 sim_327 sim_328 sim_329
0.26560606 0.16918302 0.16567303 0.19757008 0.23688133 0.14528175 0.22532543
sim_330 sim_331 sim_332 sim_333 sim_334 sim_335 sim_336
0.13139641 0.20515972 0.22472858 0.20259691 0.22442770 0.22068215 0.17894093
sim_337 sim_338 sim_339 sim_340 sim_341 sim_342 sim_343
0.20160369 0.21989302 0.25586749 0.20622181 0.20739409 0.21374384 0.17076793
sim_344 sim_345 sim_346 sim_347 sim_348 sim_349 sim_350
0.24904666 0.27743206 0.24762689 0.17258561 0.18862790 0.15727007 0.21727160
sim_351 sim_352 sim_353 sim_354 sim_355 sim_356 sim_357
0.25785456 0.17746161 0.26079682 0.23520667 0.26398817 0.19528210 0.14731120
sim_358 sim_359 sim_360 sim_361 sim_362 sim_363 sim_364
0.19143770 0.24486046 0.26935521 0.20023079 0.17103119 0.20445056 0.17705322
sim_365 sim_366 sim_367 sim_368 sim_369 sim_370 sim_371
0.21452021 0.23641234 0.25558970 0.16003569 0.26981682 0.23191497 0.16657372
sim_372 sim_373 sim_374 sim_375 sim_376 sim_377 sim_378
0.22000903 0.24068185 0.19371842 0.25963386 0.13563731 0.26118356 0.21428091
sim_379 sim_380 sim_381 sim_382 sim_383 sim_384 sim_385
0.11532751 0.21317381 0.15629030 0.17713280 0.24320152 0.19903361 0.18015394
sim_386 sim_387 sim_388 sim_389 sim_390 sim_391 sim_392
0.22555502 0.21171777 0.22056830 0.17644225 0.22345647 0.27083389 0.20695265
sim_393 sim_394 sim_395 sim_396 sim_397 sim_398 sim_399
0.18168254 0.21807963 0.26497174 0.20103321 0.24719267 0.28774290 0.30268974
sim_400 sim_401 sim_402 sim_403 sim_404 sim_405 sim_406
0.16431330 0.21958245 0.27865460 0.19940909 0.19913268 0.27102126 0.19773013
sim_407 sim_408 sim_409 sim_410 sim_411 sim_412 sim_413
0.23082389 0.24329878 0.21339022 0.19817300 0.18541989 0.17701350 0.27445250
sim_414 sim_415 sim_416 sim_417 sim_418 sim_419 sim_420
0.21966890 0.21519871 0.22163749 0.23605195 0.27479870 0.20210986 0.14122655
sim_421 sim_422 sim_423 sim_424 sim_425 sim_426 sim_427
0.19479652 0.23285486 0.26832683 0.24656040 0.29343236 0.28934047 0.19753097
sim_428 sim_429 sim_430 sim_431 sim_432 sim_433 sim_434
0.20869737 0.22660894 0.27398232 0.17475686 0.21483324 0.16846816 0.17320557
sim_435 sim_436 sim_437 sim_438 sim_439 sim_440 sim_441
0.19953703 0.20586147 0.20303399 0.22479202 0.21878546 0.24281368 0.26590961
sim_442 sim_443 sim_444 sim_445 sim_446 sim_447 sim_448
0.30895307 0.30217825 0.19211073 0.24595276 0.33664787 0.20819738 0.20526929
sim_449 sim_450 sim_451 sim_452 sim_453 sim_454 sim_455
0.23333011 0.23596861 0.19447237 0.18282777 0.20336903 0.21844044 0.25491689
sim_456 sim_457 sim_458 sim_459 sim_460 sim_461 sim_462
0.22683889 0.19801473 0.24432790 0.27238048 0.22162979 0.16552544 0.27343637
sim_463 sim_464 sim_465 sim_466 sim_467 sim_468 sim_469
0.21741088 0.20764430 0.21471712 0.19813283 0.17393387 0.34508506 0.21691541
sim_470 sim_471 sim_472 sim_473 sim_474 sim_475 sim_476
0.26756865 0.23328883 0.28471564 0.23781339 0.17928001 0.26349643 0.25802139
sim_477 sim_478 sim_479 sim_480 sim_481 sim_482 sim_483
0.22645120 0.26527855 0.17329650 0.18604797 0.26000276 0.24633672 0.18728186
sim_484 sim_485 sim_486 sim_487 sim_488 sim_489 sim_490
0.20948164 0.22759871 0.25615215 0.21814856 0.24199608 0.22670915 0.23087137
sim_491 sim_492 sim_493 sim_494 sim_495 sim_496 sim_497
0.25525735 0.25929623 0.21851712 0.28481433 0.24099708 0.16125893 0.21779922
sim_498 sim_499 sim_500 sim_501 sim_502 sim_503 sim_504
0.21304693 0.24122883 0.26040306 0.27777982 0.26498794 0.24331808 0.23892350
sim_505 sim_506 sim_507 sim_508 sim_509 sim_510 sim_511
0.27878171 0.21380687 0.25317634 0.10330095 0.26811215 0.26310252 0.22483801
sim_512 sim_513 sim_514 sim_515 sim_516 sim_517 sim_518
0.17975196 0.23155479 0.23366777 0.28598873 0.18467900 0.17993871 0.21452721
sim_519 sim_520 sim_521 sim_522 sim_523 sim_524 sim_525
0.28482233 0.19433884 0.26659559 0.16523982 0.25576432 0.17870020 0.26536325
sim_526 sim_527 sim_528 sim_529 sim_530 sim_531 sim_532
0.14581137 0.23184926 0.21550443 0.28450834 0.22184696 0.24396472 0.30390236
sim_533 sim_534 sim_535 sim_536 sim_537 sim_538 sim_539
0.15762587 0.19824633 0.20710780 0.22725603 0.24453581 0.26463633 0.21590333
sim_540 sim_541 sim_542 sim_543 sim_544 sim_545 sim_546
0.26327968 0.21520079 0.30082142 0.21459792 0.24601948 0.20916452 0.17090097
sim_547 sim_548 sim_549 sim_550 sim_551 sim_552 sim_553
0.18993414 0.13774343 0.21504332 0.20139122 0.19759526 0.23624402 0.27942475
sim_554 sim_555 sim_556 sim_557 sim_558 sim_559 sim_560
0.25009639 0.19143706 0.14401416 0.19111000 0.17739661 0.20495121 0.16323072
sim_561 sim_562 sim_563 sim_564 sim_565 sim_566 sim_567
0.29985787 0.17436651 0.25767585 0.19194740 0.23954043 0.23083522 0.24897957
sim_568 sim_569 sim_570 sim_571 sim_572 sim_573 sim_574
0.21172298 0.18617651 0.14977429 0.21345847 0.20471600 0.25514539 0.24725551
sim_575 sim_576 sim_577 sim_578 sim_579 sim_580 sim_581
0.24633498 0.24572350 0.18235894 0.28153555 0.24394770 0.20166676 0.23123742
sim_582 sim_583 sim_584 sim_585 sim_586 sim_587 sim_588
0.18614191 0.18763354 0.24340583 0.25717129 0.26890719 0.26417420 0.20332442
sim_589 sim_590 sim_591 sim_592 sim_593 sim_594 sim_595
0.15908479 0.17310929 0.22956007 0.22347350 0.19914534 0.19078601 0.25660296
sim_596 sim_597 sim_598 sim_599 sim_600 sim_601 sim_602
0.21407675 0.09915174 0.20853017 0.16912843 0.22880011 0.22125360 0.29626274
sim_603 sim_604 sim_605 sim_606 sim_607 sim_608 sim_609
0.34627590 0.24042371 0.23598849 0.21364863 0.25178582 0.26466514 0.20340968
sim_610 sim_611 sim_612 sim_613 sim_614 sim_615 sim_616
0.20718810 0.20312099 0.20955055 0.20579131 0.30534809 0.23400338 0.14119580
sim_617 sim_618 sim_619 sim_620 sim_621 sim_622 sim_623
0.24001301 0.19416576 0.27605424 0.21270662 0.22408593 0.19887095 0.18872461
sim_624 sim_625 sim_626 sim_627 sim_628 sim_629 sim_630
0.28856961 0.17731586 0.24610956 0.21508612 0.23138535 0.27266116 0.25326796
sim_631 sim_632 sim_633 sim_634 sim_635 sim_636 sim_637
0.19368895 0.23848645 0.19694181 0.22747561 0.23195704 0.21375493 0.23343966
sim_638 sim_639 sim_640 sim_641 sim_642 sim_643 sim_644
0.26289895 0.21476742 0.21044721 0.32136853 0.25351087 0.23344040 0.14275178
sim_645 sim_646 sim_647 sim_648 sim_649 sim_650 sim_651
0.25545172 0.22564473 0.20204238 0.09793875 0.24404308 0.18242491 0.22690465
sim_652 sim_653 sim_654 sim_655 sim_656 sim_657 sim_658
0.18221832 0.25120651 0.19594929 0.20048153 0.21845693 0.21342725 0.27367147
sim_659 sim_660 sim_661 sim_662 sim_663 sim_664 sim_665
0.21774304 0.22257104 0.21618034 0.22802653 0.19258848 0.29718715 0.25359780
sim_666 sim_667 sim_668 sim_669 sim_670 sim_671 sim_672
0.21830398 0.23424103 0.21124293 0.20629758 0.21543057 0.22531973 0.17900545
sim_673 sim_674 sim_675 sim_676 sim_677 sim_678 sim_679
0.21625818 0.26931981 0.22664734 0.21019805 0.14136885 0.24152259 0.19071818
sim_680 sim_681 sim_682 sim_683 sim_684 sim_685 sim_686
0.26179098 0.16692532 0.23625314 0.23968793 0.22021240 0.25200852 0.17596326
sim_687 sim_688 sim_689 sim_690 sim_691 sim_692 sim_693
0.20484537 0.19239377 0.24334572 0.22563671 0.26688518 0.22122223 0.24208963
sim_694 sim_695 sim_696 sim_697 sim_698 sim_699 sim_700
0.23189255 0.21099101 0.23446020 0.17467004 0.20444953 0.21503811 0.15836583
sim_701 sim_702 sim_703 sim_704 sim_705 sim_706 sim_707
0.22202646 0.27684460 0.20842499 0.23129594 0.13007080 0.23808299 0.26921265
sim_708 sim_709 sim_710 sim_711 sim_712 sim_713 sim_714
0.22804359 0.30200577 0.15065561 0.28217958 0.15066565 0.22411208 0.20938027
sim_715 sim_716 sim_717 sim_718 sim_719 sim_720 sim_721
0.26157566 0.24536651 0.20350043 0.17335665 0.25177002 0.21664730 0.27401579
sim_722 sim_723 sim_724 sim_725 sim_726 sim_727 sim_728
0.22134238 0.16542207 0.26983001 0.29283434 0.25513005 0.28573122 0.25084639
sim_729 sim_730 sim_731 sim_732 sim_733 sim_734 sim_735
0.21729174 0.27583701 0.25328710 0.24974066 0.20584061 0.18716906 0.19104626
sim_736 sim_737 sim_738 sim_739 sim_740 sim_741 sim_742
0.16355494 0.17318698 0.21762497 0.24951800 0.21290886 0.23618426 0.26690564
sim_743 sim_744 sim_745 sim_746 sim_747 sim_748 sim_749
0.30439584 0.25938582 0.28719305 0.22759975 0.22957353 0.28081278 0.20682834
sim_750 sim_751 sim_752 sim_753 sim_754 sim_755 sim_756
0.27227525 0.27685136 0.26287138 0.21982171 0.21650333 0.21169869 0.15706019
sim_757 sim_758 sim_759 sim_760 sim_761 sim_762 sim_763
0.21709311 0.23165096 0.26378490 0.31008802 0.23267915 0.20135886 0.22987499
sim_764 sim_765 sim_766 sim_767 sim_768 sim_769 sim_770
0.19699300 0.16331305 0.26375799 0.23995750 0.28225350 0.23035460 0.18533902
sim_771 sim_772 sim_773 sim_774 sim_775 sim_776 sim_777
0.18348215 0.24496232 0.20858452 0.29651212 0.26750853 0.26369168 0.23111811
sim_778 sim_779 sim_780 sim_781 sim_782 sim_783 sim_784
0.20383030 0.23283464 0.27131158 0.15209523 0.24652238 0.25951620 0.24224439
sim_785 sim_786 sim_787 sim_788 sim_789 sim_790 sim_791
0.29837509 0.19247686 0.21155900 0.23897237 0.21016341 0.17283805 0.20849258
sim_792 sim_793 sim_794 sim_795 sim_796 sim_797 sim_798
0.25410455 0.27077901 0.26970621 0.29389044 0.24614645 0.22436449 0.17536988
sim_799 sim_800 sim_801 sim_802 sim_803 sim_804 sim_805
0.24005082 0.25970320 0.14766385 0.22005480 0.25219694 0.20583495 0.23409499
sim_806 sim_807 sim_808 sim_809 sim_810 sim_811 sim_812
0.21502639 0.15655550 0.22878045 0.24030043 0.18067538 0.22386274 0.32769791
sim_813 sim_814 sim_815 sim_816 sim_817 sim_818 sim_819
0.22840971 0.21119578 0.21016859 0.17062218 0.24899992 0.17536063 0.28078707
sim_820 sim_821 sim_822 sim_823 sim_824 sim_825 sim_826
0.22169208 0.24140911 0.34506594 0.24599727 0.27699085 0.25973195 0.25544815
sim_827 sim_828 sim_829 sim_830 sim_831 sim_832 sim_833
0.28915518 0.19164030 0.23357880 0.18241795 0.23163730 0.20892166 0.30018131
sim_834 sim_835 sim_836 sim_837 sim_838 sim_839 sim_840
0.24330075 0.24264169 0.23152980 0.22427074 0.19229920 0.21930569 0.26368446
sim_841 sim_842 sim_843 sim_844 sim_845 sim_846 sim_847
0.20060247 0.27739951 0.24946240 0.25068789 0.22626126 0.14178086 0.10495538
sim_848 sim_849 sim_850 sim_851 sim_852 sim_853 sim_854
0.14379511 0.22742204 0.19547564 0.21994078 0.22190941 0.21718756 0.17799387
sim_855 sim_856 sim_857 sim_858 sim_859 sim_860 sim_861
0.16475724 0.22015940 0.17947279 0.28219180 0.23552952 0.32031987 0.21352569
sim_862 sim_863 sim_864 sim_865 sim_866 sim_867 sim_868
0.28327060 0.23721595 0.26968515 0.13448521 0.16160322 0.16436899 0.13267248
sim_869 sim_870 sim_871 sim_872 sim_873 sim_874 sim_875
0.23564075 0.22633150 0.28150219 0.22450230 0.25839056 0.24714680 0.15226751
sim_876 sim_877 sim_878 sim_879 sim_880 sim_881 sim_882
0.26105188 0.18665090 0.22480730 0.21834005 0.19224684 0.21537073 0.19395011
sim_883 sim_884 sim_885 sim_886 sim_887 sim_888 sim_889
0.17682835 0.17778879 0.18443464 0.20600682 0.21731572 0.20395747 0.25753473
sim_890 sim_891 sim_892 sim_893 sim_894 sim_895 sim_896
0.23349443 0.17256897 0.19725730 0.20186769 0.17405615 0.24108758 0.16371733
sim_897 sim_898 sim_899 sim_900 sim_901 sim_902 sim_903
0.19881046 0.22734189 0.27716187 0.19742679 0.25451794 0.18030984 0.20511657
sim_904 sim_905 sim_906 sim_907 sim_908 sim_909 sim_910
0.26372743 0.17316468 0.26519589 0.16176321 0.21560319 0.24343270 0.20037257
sim_911 sim_912 sim_913 sim_914 sim_915 sim_916 sim_917
0.19116068 0.22947220 0.26134865 0.28525613 0.19037585 0.23803476 0.22847206
sim_918 sim_919 sim_920 sim_921 sim_922 sim_923 sim_924
0.29180731 0.27471136 0.18720782 0.21310244 0.20876308 0.23699067 0.22396042
sim_925 sim_926 sim_927 sim_928 sim_929 sim_930 sim_931
0.24318275 0.17675095 0.21027677 0.22940603 0.24136424 0.21592415 0.16011384
sim_932 sim_933 sim_934 sim_935 sim_936 sim_937 sim_938
0.25709021 0.26619325 0.23756312 0.19059668 0.23456843 0.34184119 0.24466651
sim_939 sim_940 sim_941 sim_942 sim_943 sim_944 sim_945
0.21379225 0.20806281 0.24567435 0.18974867 0.23123240 0.22695301 0.18988286
sim_946 sim_947 sim_948 sim_949 sim_950 sim_951 sim_952
0.16857435 0.22102229 0.19992963 0.26721090 0.22765962 0.27487358 0.20501929
sim_953 sim_954 sim_955 sim_956 sim_957 sim_958 sim_959
0.19763926 0.26313933 0.25127792 0.22640643 0.16347985 0.20030193 0.22630053
sim_960 sim_961 sim_962 sim_963 sim_964 sim_965 sim_966
0.16516114 0.19058421 0.16604114 0.31122288 0.19969792 0.20354809 0.16753966
sim_967 sim_968 sim_969 sim_970 sim_971 sim_972 sim_973
0.24871500 0.21307872 0.22291048 0.23891831 0.27522200 0.22219604 0.18832574
sim_974 sim_975 sim_976 sim_977 sim_978 sim_979 sim_980
0.23179196 0.23873620 0.22065212 0.22592045 0.23937310 0.20465849 0.17676614
sim_981 sim_982 sim_983 sim_984 sim_985 sim_986 sim_987
0.20378318 0.18432447 0.22265830 0.15938733 0.23005826 0.18847937 0.24611926
sim_988 sim_989 sim_990 sim_991 sim_992 sim_993 sim_994
0.20030880 0.30898505 0.14621784 0.28304340 0.22044399 0.16953559 0.19401399
sim_995 sim_996 sim_997 sim_998 sim_999 sim_1000
0.23369440 0.33003064 0.16894296 0.22895899 0.33309221 0.24296935 # Creating a histogram
tibble(rsquared = reg_sim) |>
ggplot(aes(x = rsquared)) +
geom_histogram(bins = 30,
color = "darkorange",
fill = "orange")